Phased array planar gradient coil set for MRI systems

ABSTRACT

A uniplanar gradient coil assembly ( 40 ) generates substantially linear gradient magnetic fields through an examination region ( 14 ). The gradient coil assembly ( 40 ) includes at least a pair of primary uniplanar gradient coil sets ( 40   a   , 40   b ) and a pair of shield coil sets ( 41   a,    41   b ) which are disposed in an overlapping relationship. One gradient coil set is displaced relative to the other gradient coil set such that the mutual inductance between the two is minimized. Preferably, the coil sets ( 40   a,    40   b,    41   a,    41   b ) are symmetric, such that the sweet spot of each coil is coincident with the geometric center of each coil. One primary uniplanar gradient coil set ( 40   a ) is a high efficiency, high switching speed coil to enhance performance of ultrafast magnetic resonance sequences, while the second primary uniplanar gradient coil set ( 40   b ) is a low efficiency coil which generates a high quality gradient magnetic field, but with slower switching speeds. By displacing one gradient coil set relative to the other, mutual inductance is minimized, which maximizes peak gradient, rise time, and slew rate, while dB/dt levels are minimized. In one embodiment, the uniplanar gradient coil assembly ( 40 ) is housed within an interior of a couch ( 30 ).

BACKGROUND OF THE INVENTION

The present invention relates to the magnetic resonance arts. It findsparticular application in conjunction with gradient coils for a magneticresonance imaging apparatus and will be described with particularreference thereto. However, it is to be appreciated that the presentinvention will also find application in conjunction with localizedmagnetic resonance spectroscopy systems and other applications whichutilize gradient magnetic fields.

In magnetic resonance imaging, a uniform magnetic field is createdthrough an examination region in which a subject to be examined isdisposed. A series of radio frequency pulses and magnetic fieldgradients are applied to the examination region to excite and manipulatemagnetic resonance. Gradient fields are conventionally applied as aseries of gradient pulses with pre-selected profiles. The radiofrequency pulses excite magnetic resonance and the gradient field pulsesphase and frequency encode the induced resonance. In this manner, phaseand frequency encoded magnetic resonance signals are generated.

More specifically, the gradient magnetic field pulses are typicallyapplied to select and encode the magnetic resonance with spatialposition. In some embodiments, the magnetic field gradients are appliedto select a slice or slab to be imaged. Ideally, the phase or frequencyencoding uniquely identifies spatial location.

Conventionally, the uniform main magnetic field is generated in one oftwo ways. The first method employs a cylindrically shaped solenoidalmain magnet. The central bore of the main magnet defines the examinationregion in which a horizontally directed main magnetic field isgenerated. The second method employs a main magnet having opposing polesarranged facing one another to define therebetween the examinationregion. The poles are typically connected by a C-shaped or a four postferrous flux return path. This configuration generates a verticallydirected main magnetic field within the examination region. The C-shapedmain magnet, often referred to as having open magnet geometry, hasresolved important MRI issues, such as increasing the patient aperture,avoiding patient claustrophobia, and improving access for interventionalMRI applications. However, the design of gradient coils for generatinglinear magnetic field gradients differs from that for the bore-typehorizontal field system due to the direction of the magnetic field.

When designing gradient coils for magnetic resonance imaging, manyopposing factors must be considered. Typically, there is a trade offbetween gradient speed and image quality factors, such as volume,uniformity, and linearity. Some magnetic resonance sequences require agradient coil which emphasizes efficiency, while other sequences arebest with a gradient coil which emphasizes image quality factors. Forexample, a gradient coil which has a large linear imaging volume isadvantageous for spine imaging, but is disadvantageous in terms of thedB/dt when switched with a high slew rate.

Open magnetic systems with vertically directed fields are attractive forMRI applications because an open magnet geometry increases the patientaperture and increases access for interventional MRI applications. Insuch open magnet systems, it has been known to use a bi-planar gradientcoil assembly for generation of magnetic field gradients. However, theuse of this type of bi-planar gradient coil assembly somewhat detractsfrom the purposes for using an open magnet geometry in that it reducesthe patient aperture and diminishes access for interventional proceduresby employing two planar gradient coils, one on either side of thesubject being examined. In addition, the performance of the bi-planarconfiguration often suffers in terms of its gradient strength, slewrate, and dB/dt levels.

A single uniplanar gradient coil may remedy some of the aforementionedbi-planar shortcomings in regard to gradient strength and slew rate.However, such a structure often suffers from a reduced field of view,which affects applications requiring larger spatial coverage, such asspinal imaging. In order to increase the spatial coverage provided by auniplanar or bi-planar gradient coil, the uniformity and linearity ofthe gradient magnetic field must be improved. In addition, the strengthof the gradient field must be increased to address the demand for ahigher resolution image. These two factors have a detrimental effect onthe dB/dt level for the gradient coil.

The present invention contemplates a new and improved gradient coilassembly which overcomes the above-referenced problems and others.

SUMMARY OF THE INVENTION

In accordance with one aspect of the present invention, a magneticresonance imaging apparatus includes a main magnet for generating a mainmagnetic field through and surrounding an examination region. A couchsupports a subject within the examination region. A planar gradient coilassembly is disposed on one side of the subject and generates gradientmagnetic fields across the examination region. The planar gradient coilassembly includes at least a first primary planar gradient coil set anda second primary planar gradient coil set which are disposed in anoverlapping relationship. The first primary planar gradient coil set isdisplaced relative to the second primary planar gradient coil set suchthat mutual inductance between the two planar gradient coil sets isminimized. A current supply supplies electrical current to the planargradient coil assembly such that magnetic field gradients areselectively generated across the examination region in the main magneticfield by the planar gradient coil assembly. An RF pulse generatorselectively excites magnetic resonance dipoles disposed within theexamination region and a receiver receives magnetic resonance signalsfrom resonating dipoles within the examination region. A reconstructionprocessor reconstructs the demodulated magnetic resonance signals intoan image representation.

In accordance with another aspect of the present invention, a method ofmagnetic resonance includes generating a vertical main magnetic fieldacross an examination region. It further includes applying a firstgradient magnetic field across the examination region with a firstplanar gradient coil during resonance excitation. A second gradientmagnetic field is applied across the examination region with a secondplanar gradient coil during resonance data acquisition. The acquiredresonance data is reconstructed into an image representation.

In accordance with another aspect of the present invention, a method ofdesigning a phased array gradient coil assembly for a magnetic resonanceimaging system includes selecting geometric configurations for a primarycoil set having a corresponding shield coil set and a second primarycoil set having a corresponding second shield coil set. The methodfurther includes generating first and second continuous currentdistributions for the first primary and shield coil sets and third andfourth continuous current distributions for the second primary andshield coil sets. The first primary coil set is optimized with the firstshield coil set using an energy/inductance minimization algorithm. Next,the second primary coil set is optimized with the second shield coil setusing an energy/inductance minimization algorithm. Eddy currents areevaluated within the prescribed imaging volume for both the first andsecond primary coil sets and at least one characteristic of thegeometric configurations defined above are modified if the eddy currentsdo not meet specified target values. The first primary and shield coilsets and the second primary and shield coil sets are discretized. Thefirst primary coil set is at least one of axially and radially displacedrelative to the second primary coil set such that mutual inductancebetween the two is minimized.

In accordance with another aspect of the present invention, a phasedarray planar gradient coil assembly for generating magnetic gradientsacross a main magnetic field of a magnetic resonance apparatus includesa first primary planar coil set and a second primary planar coil seteach including first and second planar x, y, and z-gradient coils havingsweet spots in which the magnetic field gradient generated issubstantially linear. The respective sweet spots are coincident with anexamination region. The first and second primary planar coil sets aredisposed in an overlapping relationship with the first primary planarcoil set being displaced relative to the second primary planar coil setsuch that the mutual inductance between the two is minimized. Thegradient coil assembly further includes at least one planar shieldingcoil set between the first and second primary planar coil sets and amain magnet. The at least one planar shielding coil set generates amagnetic field which substantially zeroes magnetic field gradientsoutside the shielding coil set.

One advantage of the present invention is that it reduces the resistanceof the gradient coil assembly.

Another advantage of the present invention is that it increases the dutycycle of the gradient coil assembly.

Another advantage of the present invention is that it minimizes mutualinductance of the gradient coil assembly.

Another advantage of the present invention is that it increases thefield of view of the system without increasing the energy of the system.

Yet another advantage of the present invention is that it reduces dB/dtlevels.

Another advantage of the present invention is that it minimizes torqueand thrust forces.

Still further advantages and benefits of the present invention willbecome apparent to those of ordinary skill in the art upon reading andunderstanding the following detailed description of the preferredembodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention may take form in various components and arrangements ofcomponents, and in various steps and arrangements of steps. The drawingsare only for purposes of illustrating preferred embodiments and are notto be construed as limiting the invention.

FIG. 1 is a diagrammatic illustration of a magnetic resonance imagingsystem in accordance with the present invention;

FIG. 2 is a diagrammatic illustration of the planar gradient coilassembly in accordance with the present invention;

FIG. 3 is a flow chart for designing a shielded planar gradient coilassembly with minimized mutual inductance in accordance with the presentinvention;

FIGS. 4A and 4B are diagrammatic illustrations of an exemplary firstprimary symmetric uniplanar z-gradient coil and a first shield coil fora vertical system in accordance with the present invention;

FIG. 5 is a plot of the y-component of the magnetic field along the yzplane at x=0.0 for the first shielded z uniplanar gradient coil for avertical system in accordance with the present invention;

FIGS. 6A and 6B are diagrammatic illustrations of an exemplary secondprimary symmetric uniplanar z-gradient coil and a second shield coil fora vertical system in accordance with the present invention;

FIG. 7 is a plot of the y-component of the magnetic field along the yzplane at x=0.0 for the second shielded z uniplanar gradient coil for avertical system in accordance with the present invention;

FIG. 8 is a plot of mutual energy vs. z-shift for the first and secondsymmetric transverse uniplanar gradient coils for a vertical system inaccordance with the present invention;

FIGS. 9A and 9B are diagrammatic illustrations of an exemplary firstprimary symmetric uniplanar y-gradient coil and a corresponding firstshield coil for a vertical system in accordance with the presentinvention;

FIGS. 10A and 10B are diagrammatic illustrations of an exemplary secondprimary symmetric uniplanar y-gradient coil and a corresponding secondshield coil for a vertical system in accordance with the presentinvention;

FIG. 11 is a plot of mutual energy vs. z-shift for the first and secondsymmetric uniplanar y-gradient coils for a vertical system in accordancewith the present invention;

FIGS. 12A and 12B are diagrammatic illustrations of an exemplary firstprimary symmetric uniplanar z-gradient coil and a first shield coil fora horizontal system in accordance with the present invention;

FIGS. 13A and 13B are diagrammatic illustrations of an exemplary secondprimary symmetric uniplanar z-gradient coil and a second shield coil fora horizontal system in accordance with the present invention;

FIG. 14 is a plot of mutual energy vs. z-shift for the first and secondsymmetric uniplanar z-gradient coils for a horizontal system inaccordance with the present invention;

FIGS. 15A and 15B are diagrammatic illustrations of an exemplary firstprimary symmetric uniplanar y-gradient coil and a first shield coil fora horizontal system in accordance with the present invention;

FIGS. 16A and 16B are diagrammatic illustrations of an exemplary secondprimary symmetric uniplanar y-gradient coil and a second shield coil fora horizontal system in accordance with the present invention;

FIG. 17 is a plot of mutual energy vs. z-shift for the first and secondsymmetric uniplanar y-gradient coils for a horizontal system inaccordance with the present invention; and

FIGS. 18A, 18B, and 18C are diagrammatic illustrations of variousconfigurations for connecting the first and second uniplanar gradientcoil sets for selective excitation in accordance with the presentinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

With reference to FIG. 1, a magnetic resonance imaging apparatus 10generates a substantially uniform vertical magnetic field 12 through anexamination region 14 defined by opposing magnetic pole pieces 16, 18.Preferably, the main magnetic field 12 is a strong, substantiallyuniform field that is aligned with a y or vertical axis. Although anopen field magnet with a vertically directed field is illustrated inFIG. 1, it is to be appreciated that the present invention is equallyapplicable to bore-type magnetic systems with horizontally directedfields (along the z-direction). In the illustrated embodiment, theopposing magnetic pole pieces 16, 18 are connected by a ferrous fluxreturn path 20. Electrical main field coils 22 are operated undercontrol of a main magnetic field control circuit 24. Preferably, themain magnetic field coils 22 include coil segments disposed adjacent toor in conjunction with each of the opposing magnetic pole pieces 16, 18.However, the main field coils 22 may be disposed anywhere along theferrous flux return path 20.

A couch 30 suspends a subject within the examination region 14.Preferably, the couch 30 is movable so as to be selectively inserted andretracted from the examination region. An interior cavity 32 of thecouch 30 houses a uniplanar gradient coil assembly 40. The uniplanargradient coil assembly 40 selectively creates gradient magnetic fieldsacross the examination region 14. In one embodiment, a mechanicaladjustment mechanism 70 adjusts the height of the uniplanar gradientcoil assembly 40 to align the magnetic gradient region having optimumlinearity with an area of interest of the subject.

A current supply 42 selectively supplies electrical current pulses tothe coil loop arrays of the uniplanar gradient coil assembly 40. Agradient field control means 44 is controlled by a sequence controlprocessor 46 to control the current supply to apply appropriate currentpulses to the windings of the coil loop arrays to cause selectedgradient pulses.

The sequence control processor 46 controls a radio frequency transmitter50 for generating radio frequency pulses of the appropriate frequencyspectrum for inducing resonance in selected dipoles disposed in theexamination region 14. The radio frequency transmitter 50 is connectedto a radio frequency antennae 52 disposed adjacent the examinationregion for transmitting radio frequency pulses into a region of interestof the patient or other subject in the examination region 14. The radiofrequency antennae may be disposed adjacent a surface of the magneticpole pieces 16, 18, in the interior cavity 32 of the couch 30, or on thesubject to be examined. For example, a surface coil may be positionedcontiguous to an examined patient or subject for controllably inducingmagnetic resonance in a selected contiguous region of the patient.

A magnetic resonance receiver 54 receives signals from resonatingdipoles within the examination region 14. The signals are received viathe same antennae that transmits the radio frequency pulses.Alternately, separate receiver coils may be used. For example, receiveonly surface coils may be disposed contiguous to a selected region ofthe patient to receive resonance induced therein by a radio frequencytransmitting coil surrounding the examination region 14. Ultimately, theradio frequency signals received are demodulated and reconstructed intoan image representation by a reconstruction processor 62. The image mayrepresent a planar slice through the patient, an array of parallelplanar slices, a three dimensional volume, or the like. The image isthen stored in an image memory 64 where it may be accessed by a display66, such as a video monitor, which provides a human-readable display ofthe resultant image.

With reference to FIG. 2 and continuing reference to FIG. 1, thegradient coil assembly preferably includes a pair of primary gradientcoil sets 40 a, 40 b and a corresponding pair of shield coil sets 41 a,41 b. In one embodiment, the gradient coil assembly 40 is housed withinthe couch 30. Alternately, the gradient coil assembly may be locatedoutside of the couch on one side of the subject.

Preferably, the primary gradient coil sets are symmetric, i.e., eachcoil set's “sweet spot” (region where the gradient magnetic field issubstantially linear) is coincident with the geometric center of thegradient coil set. It is to be appreciated that the present inventionmay contain only asymmetric gradient coil sets, only symmetric gradientcoil sets, and both asymmetric and symmetric gradient coil sets. Inaddition, it is to be appreciated that more than two primary gradientcoil sets may be overlapped in a phased array relationship. Such aphased array configuration increases the field of view for applicationssuch as spinal imaging. Also, it is to be appreciated that there is norequirement that the sweet spots of the respective gradient coil setscoincide. Further, it is to be appreciated that two uniplanar phasedarrays may be symmetrically disposed on both sides of the examinationregion to form a bi-polar phased array gradient coil assembly.

As shown in FIG. 2, the first primary gradient coil set 40 a isdisplaced axially (i.e., along the z-axis) with respect to the secondprimary gradient coil set 40 b. Alternately, the first primary gradientcoil set may be displaced axially, i.e., with respect (along the y-axis)to the second primary gradient coil set. The axial and/or radialdisplacement of one primary gradient coil set relative to the other isdetermined by a mutual inductance minimization algorithm, which will bediscussed in full detail below. By displacing one gradient coil setrelative to another gradient coil set, the mutual inductance orinteraction between the two is minimized. For example, in FIG. 2, thesecond primary gradient coil 40 b is displaced axially (along the zdirection) by an amount Z_(SHIFT). This feature maximizes peak gradient,rise time, and slew rate and leads to overall greater coil efficiency.Further, using uniplanar gradient coils in the inductance minimizingconfiguration lowers the coil's resistance and increases the duty cycleof the coil. With uniplanar coils, the entire length of the coil isutilized which controls the current density by allowing it to bedistributed with wider copper loops for lower resistance and reducedheat dissipation. As a result, no elaborate cooling system is required.

Preferably, the first primary gradient coil set is a high efficiencycoil set which enhances performance of ultrafast magnetic resonancesequences. Such a coil set minimizes dB/dt levels and eddy currenteffects. Preferably, the second primary gradient coil set is a lowefficiency coil set capable of generating a high quality gradient field.Such a coil set typically is ideal for imaging applications withinherently low dB/dt and eddy current levels.

Each uniplanar gradient coil set of the uniplanar gradient coil assemblyincludes an x, y, and z coil loop array. The y-gradient coil loop arrayapplies gradients along a y-axis. Analogously, the x and z-gradient coilloop arrays generate gradients along the x and z-axes, respectively.Each of the x, y, and z-gradient coil loop arrays include a plurality ofsymmetrically arranged windings or coil loops, as shown in the figuresbelow. Each of the coil loop arrays are disposed in an individual planarsurface which is orthogonal to the main magnetic field 12 in a verticalfield system. Alternately, in a bore-type magnet, the coil loop arraysare parallel to the main horizontal magnetic field. The windings arepreferably manufactured from a relatively thin conductive sheet, such ascopper. The sheet is preferably cut before lamination to the former bywater jet cutting, laser cutting, etching or the like, and then bondedto a thin insulating substrate, minimizing radial thickness.

The theoretical development, the design procedure and the numericalresults for two symmetric, shielded uniplanar gradient coil sets, eachconsisting of three gradients coils, is now discussed. In addition, thetheoretical development and numerical results for two sets of activelyshielded symmetric gradient coils with minimized mutual inductance isdiscussed. Specifically, the theoretical development, the design, andthe results of a gradient coil where the y component of the magneticfield varies linearly along the transverse direction (z, z-gradientcoil), as well as, the axial gradient coil (y, y-gradient coil) will bepresented.

The flow chart for designing such a gradient coil assembly is shown inFIG. 3. Initially, a geometric configurations step sets the geometricconfigurations of the first uniplanar primary coil with the firstsymmetric shield 100 and the second uniplanar primary coil with thesecond symmetric shield 102. Next, an energy/inductance minimizationstep 104 optimizes each gradient coil set. In addition, a torqueminimization step 106 minimizes the torque on each gradient coil set.Next, eddy current inside a prescribed imaging volume is evaluated foreach primary coil configuration 108. The eddy currents from each primarycoil set with their corresponding shields are compared to target valuesfor the particular volume 110.

If the eddy current target values are met, a current discretization step112 discretizes the continuous current distributions of each coil set togenerate the number of turns which is required for each coil within eachcoil set. Next, the primary coil and associated shield are examined todetermine whether each has an exact integer number of turns when theyshare common current 112. If this condition is not satisfied, the fieldcharacteristics and/or coil geometric configurations are modified 114and the process proceeds again from the optimization step 104.Similarly, if the eddy current inside the prescribed imaging volume 110do not meet the target values, the field characteristics and/or coilgeometric configurations are modified 116 and the process proceeds againfrom the optimization step 104. The process continues until anacceptable solution is found 118 which satisfies the target criteria.

The theoretical development of the energy optimization algorithm step104 is discussed for both the transverse and the axial gradient coil.

Considering the plane for the primary coil positioned at y=−a, while theplane for the secondary coil is positioned at y=−b, and confining thecurrent density on the xz plane, the expression for the currentdistribution J(x,z) is:

{right arrow over (J)}(x,z)=[J _(x)(x,z){circumflex over (x)}+J_(z)(x,z){circumflex over (z)}]  (1).

Therefore, the expressions of the two components A_(x), A_(z) of themagnetic vector potential for y≧−a are: $\begin{matrix}{A_{x} = {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {\text{~~~~~~~~~}\text{~~~~~~~~~~~~~~~~~~~~~~~~}{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right\rbrack}}}}}} & (2) \\{A_{z} = {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack {{{J_{z}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {\text{~~~~~~~~~~}\text{~~~~~~~~~~~~~~~~~~~~~~~~}{J_{z}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right\rbrack}}}}}} & (3)\end{matrix}$

In addition, the expressions of the two components A_(x), A_(z) of themagnetic vector potential for y≦−b are: $\begin{matrix}{A_{x} = {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}\quad {^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {\text{~~~~~~~~~~~}\text{~~~~~~~~~~~~~~~~~~~~~~~~~~}{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack}}}}}} & (4) \\{A_{z} = {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack {{{J_{z}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {\text{~~~~~~~~~~~}\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}{J_{z}^{- b}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack}}}}}} & (5)\end{matrix}$

where J_({x,z})(α,β) represents the double Fourier transform of theJ_({x,z})(x,z) components of the current density, respectively.Considering the current continuity equation ({overscore (V)}·J=0), therelationship for these two components of the current density in theFourier domain is: $\begin{matrix}{{J_{z}\left( {\alpha,\beta} \right)} = {{- \frac{\alpha}{\beta}}{J_{x}\left( {\alpha,\beta} \right)}}} & (6)\end{matrix}$

In this case, Equations (3) and (5) have the form: $\begin{matrix}{A_{z} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( \quad {- \frac{\alpha}{\beta}} \right)}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right\rbrack}\quad {for}\quad y}}}} \geq {- a}}} & (7) \\{A_{z} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( \quad {- \frac{\alpha}{\beta}} \right)}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {{J_{x}^{- b}\left( \quad {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack}\quad {for}\quad y}}}} \leq {- b}}} & (8)\end{matrix}$

Furthermore, the expressions of the two components for magnetic vectorpotential at the surfaces of the two planes (y=−a, and y=−b) have theform: $\begin{matrix}{A_{x} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\quad \frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack {{J_{x}^{- a}\left( {\alpha,\beta} \right)} + {\text{~~~~~~~~~~~~~~}\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~}{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}}} \right\rbrack}\quad {for}\quad y}}}} = {- a}}} & (9) \\{A_{z} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\quad \frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \frac{\alpha}{\beta}} \right)}\left\lbrack {{J_{x}^{- a}\left( {\alpha,\beta} \right)} + {\text{~~~~~~~~~~~~}\text{~~~~~~~~~~~~~~~~~~~}{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}}} \right\rbrack}\quad {for}\quad y}}}} = {- a}}} & (10) \\{A_{x} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\quad \frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}} + {\text{~~~~~~~}\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}{J_{x}^{- b}\left( {\alpha,\beta} \right)}}} \right\rbrack}\quad {for}\quad y}}}} = {- b}}} & (11) \\{A_{z} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\quad \frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \frac{\alpha}{\beta}} \right)}\left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}} + {\text{~~~~~~~~~~~~~~~~~~~~~~~~~~}{J_{x}^{- b}\left( {\alpha,\beta} \right)}}} \right\rbrack}\quad {for}\quad y}}}} = {- b}}} & (12)\end{matrix}$

The expression of the component of the gradient field coincidental withthe direction of the main magnetic field is B_(y), and has the followingform: $\begin{matrix}{B_{y} = {\left. {{\partial_{z}A_{x}} - {\partial_{x}A_{z}}}\Rightarrow B_{y} \right. = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{{\frac{\quad {{\alpha}{\beta}}}{\sqrt{\alpha^{2} + \beta^{2}}}\left\lbrack \quad {{\quad \beta} + {\frac{\alpha^{2}}{\beta}}} \right\rbrack}\quad {^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack \quad {{{J_{x}^{- a}\left( \quad {\alpha,\quad \beta} \right)}\quad ^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right\rbrack}\quad {for}\quad y}}}} \geq {- a}}}} & (13) \\{B_{y} = {{\frac{\mu_{0}}{8\pi^{2}}\quad {\int{\int_{- \infty}^{+ \infty}{{\frac{\quad {{\alpha}{\beta}}}{\sqrt{\alpha^{2} + \beta^{2}}}\left\lbrack \quad {{\quad \beta} + {\frac{\alpha^{2}}{\beta}}} \right\rbrack}\quad {^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {\text{~~~~~~~~~~~~~~~~}{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack}\quad {for}\quad y}}}} \leq {- b}}} & (14)\end{matrix}$

Considering the shielding requirements B_(y)=0 at y=−b, Equation (14)relates the current densities for the two planes as: $\begin{matrix}{{J_{x}^{- b}\left( {\alpha,\beta} \right)} = {{- ^{\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}}{J_{x}^{- a}\left( {\alpha,\beta} \right)}}} & (15)\end{matrix}$

and Equation (13) becomes: $\begin{matrix}{B_{y} = {{\frac{\quad \mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\quad {\alpha}{\beta}\frac{\sqrt{\alpha^{2} + \beta^{2}}}{\beta}^{{\quad \alpha \quad x} + {\quad \beta \quad z}}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}{{J_{x}^{- a}\left( {\alpha,\beta} \right)}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}\quad {for}\quad y}}}} \geq {- a}}} & (16)\end{matrix}$

In addition, the stored magnetic energy of the coil is given by:$\begin{matrix}{\quad {W_{m} = {\left. {\frac{1}{2}{\int_{v}{{^{3}x}\quad {\overset{\rightarrow}{A} \cdot \overset{\rightarrow}{J}}}}}\Rightarrow W_{m} \right. = {\frac{\mu_{0}}{16\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {{\alpha}{\beta}\frac{\sqrt{\alpha^{2} + \beta^{2}}}{\beta^{2}}{{{J_{x}^{- a}\left( {\alpha,\beta} \right)}}^{2}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}}}}}}}} & (17)\end{matrix}$

In this section, the theoretical development for the transverseuniplanar shielded gradient coil will be presented. Specifically, the Xgradient coil will be presented in its entirety, while the Z gradientcoil is simply a rotation of the X current patterns by 90° with respectto the y axis.

For the X gradient coil, the y component of the gradient field must beantisymmetric along the x direction and symmetric along the z and ydirections. In this case, Equation (16) becomes: $\begin{matrix}{{B_{y} = {{{- \frac{\mu_{0}}{8\pi^{2}}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}\quad {\beta}\frac{\sqrt{\alpha^{2} + \beta^{2}}}{\beta}{\sin \left( {\alpha \quad x} \right)}{\cos\left( \quad {\beta \quad z} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}{{J_{x}^{- a}\left( \quad {\alpha,\quad \beta} \right)}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}\quad {for}\quad y}}}} \geq {- a}}}\quad} & (18)\end{matrix}$

which leads to the expression of the Fourier transform for the J_(x)^(−a)(α,β) as: $\begin{matrix}{{J_{x}^{- a}\left( {\alpha,\beta} \right)} = {{- 4}{\int{\int_{0}^{+ \infty}\quad {{x}{z}\quad {\sin \left( {\alpha \quad x} \right)}{\sin \left( {\beta \quad z} \right)}{J_{x}^{- a}\left( {x,z} \right)}}}}}} & (19)\end{matrix}$

Based on the energy minimization mechanism, the functional E isconstructed in terms of the magnetic field and the stored magneticenergy as: $\begin{matrix}{{E\left( J_{x}^{- a} \right)} = {W_{m} - {\sum\limits_{j = 1}^{N}{\lambda_{j}\left\lbrack {{B_{y}\left( {\overset{\rightarrow}{r}}_{j} \right)} - {B_{ySC}\left( {\overset{\rightarrow}{r}}_{j} \right)}} \right\rbrack}}}} & (20)\end{matrix}$

where λ_(j) are the Lagrange multipliers, B_(y)(r_(j)) is the calculatedvalue of the magnetic field at the constraint points r_(j) andB_(ySC)(r_(j)) are the constraint values of the magnetic field at theconstraint points.

Minimizing E with respect to the current density J_(x) ^(−a), a matrixEquation for J_(x) ^(−a) is: $\begin{matrix}{J_{x}^{- a} = {{- \beta}{\sum\limits_{j = 1}^{N}{\lambda_{j}{\sin \left( {\alpha \quad x_{j}} \right)}{\cos \left( {\beta \quad z_{j}} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}}} & \text{(20a)}\end{matrix}$

where the Lagrange multipliers λ_(j) are determined via the constraintEquation: $\begin{matrix}{{{\sum\limits_{j = 1}^{N}{C_{ij}\lambda_{j}}} = {B_{{ySC}_{i}}\quad {with}}}{C_{ij} = {\frac{\mu_{0}}{2\pi^{2}}{\int{\int_{0}^{+ \infty}\quad {{\alpha}{\beta}\sqrt{\alpha^{2} + \beta^{2}}{\sin \left( {\alpha \quad x_{i}} \right)}{\cos \left( {\beta \quad z_{i}} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{i} + a})}}}}}}}{\sum\limits_{j = 1}^{N}{\lambda_{j}{\sin \left( {\alpha \quad x_{j}} \right)}{\cos \left( {\beta \quad z_{j}} \right)}{^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}}}} & (21)\end{matrix}$

Upon determination of the Lagrange multipliers, the continuous currentdensity distribution for J_(x) ^(−a) and J_(z) ^(−a) components of thecurrent density are generated. For the shielding coil, its currentdensity can be derived by determining the current density of the primarycoil via Equation (20a) and subsequently applying the inverse transformto Equation (15). Upon determination of the continuous current densitiesfor both coils, their associated discrete current patterns, which arevery close approximations for the continuous current densities can begenerated by applying the stream function technique. In order to ensurethe integrity of the discretization process, the magnetic field isre-evaluated inside and outside the imaging volume by applying theBiot-Savart formula to both discrete current distributions.

In this section, the theoretical development for the axial uniplanarshielded gradient coil will be presented.

For the Y gradient coil, the y component of the gradient field must besymmetric along the x and z directions. In this case, Equation (16)becomes: $\begin{matrix}{B_{y} = {{{- \frac{\mu_{0}}{8\pi^{2}}}{\int{\int_{- \infty}^{+ \infty}{\quad {\alpha}{\beta}\frac{\sqrt{\alpha^{2} + \beta^{2}}}{\beta}{\cos \left( {\alpha \quad x} \right)}{\cos \left( {\beta \quad z} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}{{{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}\quad {for}\quad y}}}} \geq {- a}}} & (22)\end{matrix}$

which leads to the expression of the Fourier transform for the J_(x)^(−a)(α,β) as: $\begin{matrix}{{{J_{x}^{- a}\left( {\alpha,\beta} \right)} = {\quad {{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)}}}{and}{{{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)} = {4{\int{\int_{0}^{+ \infty}\quad {{x}{z}\quad {\cos \left( {\alpha \quad x} \right)}{\sin \left( {\beta \quad z} \right)}{J_{x}^{- a}\left( {x,z} \right)}}}}}}} & (23)\end{matrix}$

Based on the energy minimization mechanism, the functional E isconstructed in terms of the magnetic field and the stored magneticenergy as: $\begin{matrix}{{E\left( {\hat{J}}_{x}^{- a} \right)} = {W_{m} - {\sum\limits_{j = 1}^{N}{\lambda_{j}\left\lbrack {{B_{z}\left( {\overset{\rightarrow}{r}}_{j} \right)} - {B_{zSC}\left( {\overset{\rightarrow}{r}}_{j} \right)}} \right\rbrack}}}} & (24)\end{matrix}$

where λ_(j) are the Lagrange multipliers, B_(y)(r_(j)) is the calculatedvalue of the magnetic field at the constraint points r_(j) andB_(ySC)(r_(j)) are the constraint values of the magnetic field at theconstraint points.

Minimizing E with respect to the current density Ĵ _(x) ^(−a), a matrixEquation for Ĵ _(x) ^(−a) is: $\begin{matrix}{{\hat{J}}_{x}^{- a} = {{- \beta}{\sum\limits_{j = 1}^{N}{\lambda_{j}{\cos \left( {\alpha \quad x_{j}} \right)}{\cos \left( {\beta \quad z_{j}} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}}} & \text{(24a)}\end{matrix}$

where the Lagrange multipliers λ_(j) are determined via the constraintEquation: $\begin{matrix}{{{\sum\limits_{j = 1}^{N}{C_{ij}\lambda_{j}}} = {B_{{ySC}_{i}}\quad {with}}}\begin{matrix}{C_{ij} = \quad {\frac{\mu_{0}}{2\pi^{2}}{\int{\int_{0}^{+ \infty}\quad {{\alpha}{\beta}\sqrt{\alpha^{2} + \beta^{2}}{\cos \left( {\alpha \quad x_{i}} \right)}{\cos \left( {\beta \quad z_{i}} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{i} + a})}}}}}}} \\{\quad {\sum\limits_{j = 1}^{N}{\lambda_{j}{\cos \left( {\alpha \quad x_{j}} \right)}{\cos \left( {\beta \quad z_{j}} \right)}{^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}\quad\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}}}}\end{matrix}} & (25)\end{matrix}$

Upon determination of the Lagrange multipliers the continuous currentdensity distribution for J_(x) ^(−a) and J_(z) ^(−a) components of thecurrent density are generated. For the shielding coil, its currentdensity can be derived by determining the current density of the primarycoil via Equation (24a) and subsequently applying the inverse transformto Equation (15). Upon determination of the continuous current densitiesfor both coils, their associated discrete current patterns, which arevery close approximations for the continuous current densities can begenerated by applying the stream function technique. In order to ensurethe integrity of the discretization process, the magnetic field isre-evaluated inside and outside the imaging volume by applying theBiot-Savart formula to both discrete current distributions.

Considering two sets of shielded uniplanar gradient modules, where forthe first module the plane positions for the primary and secondary coilsare y=−a′, and y=−b′, respectively, while the plane locations for theprimary and secondary coils of the second module are y=−a and y=−brespectively. In this example, for computing the mutual energy of thecoils, −a<−a′ and −b>−b′. Also it is considered that the first modulecoil is shifted axially along the z direction by z₀. In this situation,the Fourier transform of the current density for the first module coilcan be written as:

J _(x,z) ^(−a′,−b′)(α,β)=e ^(−iβz) ^(₀) ∫∫_(−∞) ^(+∞) e ^(−iαx′−iβ(z′−z)^(₀) ⁾ J _(x,z) ^(−a′,−b′)(x′,z′)  (26)

In addition to Equations (9)-(13), the expressions of the vectorpotential at the locations of the first module are $\begin{matrix}{A_{x} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({a - a^{\prime}})}}} + {{J_{x}^{- b}\left( {\alpha,\quad \beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a^{\prime}})}}}} \right\rbrack}\quad {for}\quad y}}}} = {- a^{\prime}}}} & (27) \\{{A_{z} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \frac{\alpha}{\beta}} \right)}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({a - a^{\prime}})}}} + \quad {{J_{x}^{- b}\left( \quad {\alpha,\quad \beta} \right)}\quad ^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a^{\prime}})}}}} \right\rbrack}\quad {for}\quad y}}}} = {- a^{\prime}}}}\quad} & (28) \\{A_{x} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b^{\prime} - a})}}} + {{J_{x}^{- b}\left( {\alpha,\quad \beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - b^{\prime}})}}}} \right\rbrack}\quad {for}\quad y}}}} = {- b^{\prime}}}} & (29) \\{A_{z} = {{\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \frac{\alpha}{\beta}} \right)}\left\lbrack \quad {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - b^{\prime}})}}}} \right\rbrack}\quad {for}\quad y}}}} = {- b^{\prime}}}} & (30)\end{matrix}$

The expression for the mutual energy between these two modules is:

W _(mutual)=½∫_(v) {right arrow over (A)} ^((−a,−b)) · {right arrow over(J)} ^((−a′,−b′))

and with the help of equations (25)-(30), it becomes: $\begin{matrix}{W_{mutual} = {\frac{\mu_{0}}{4\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {{\alpha}{\beta}\frac{\sqrt{\alpha^{2} + \beta^{2}}}{\beta^{2}}^{\quad \beta \quad z_{0}}{J_{x}^{- a}\left( {\alpha,\beta} \right)}{J_{x}^{- a^{\prime*}}\left( {\alpha,\beta} \right)}{^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({a - a^{\prime}})}}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - {({a - a^{\prime}})}})}}} \right\rbrack}}}}}} & (31)\end{matrix}$

If the mutual energy is evaluated over two gradient modules whichrepresent the same gradient axis and the current density represents thesame axis coil, the above expression becomes: $\begin{matrix}{W_{mutual} = {\frac{\mu_{0}}{4\pi^{2}}{\int{\int_{- \infty}^{+ \infty}\quad {{\alpha}{\beta}\frac{\sqrt{\alpha^{2} + \beta^{2}}}{\beta^{2}}\cos \quad \left( {\beta \quad z_{0}} \right){J_{x}^{- a}\left( {\alpha,\beta} \right)}{J_{x}^{- a^{\prime*}}\left( {\alpha,\beta} \right)}{^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({a - a^{\prime}})}}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - {({a - a^{\prime}})}})}}} \right\rbrack}}}}}} & (32)\end{matrix}$

For the first shielded Z uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a′=−0.002 m andy=−b′=−0.056 m, respectively. Six constraint points were chosen todefine the characteristics of the field inside an imaging volume. Thefirst three constraint points define a gradient strength of 20 mT/minside the imaging volume with a 12% on-axis linearity. The reason ofusing three constraint points along the gradient axis is that theon-axis non-linearity of the gradient coil must be contained inacceptable levels. The uniformity of the gradient field inside thisimaging volume is restricted to less than 17% from its actual value.This set of constraints is displayed in Table 1.

TABLE 1 Constraints for the first Z shielded Uniplanar Module j X_(j) inmm Y_(j) in mm Z_(j) in mm B_(ysc) in mTesla 1 0.0 100.0 1.000 0.0200 20.0 100.0 100.0 2.0000 3 0.0 100.0 320.0 7.6400 4 180.0 100.0 1.0000.0190 5 0.0 150.0 1.000 0.0195 6 0.0 200.0 1.000 0.0165

For the second shielded Z uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a=−0.006 m andy=−b=−0.050 m, respectively. Six constraint points were chosen to definethe characteristics of the field inside an imaging volume. The firstthree constraint points define a gradient strength of 20 mT/m inside theimaging volume with a 15% on-axis linearity. The reason for using threeconstraint points along the gradient axis is that the on-axisnon-linearity of the gradient coil must be contained in acceptablelevels. The uniformity of the gradient field inside this imaging volumeis restricted to less than 20% from its actual value. This set ofconstraints is displayed in Table 2.

TABLE 2 Constraints for the second Z shielded Uniplanar module j X_(j)in mm Y_(j) in mm Z_(j) in mm B_(ysc) in mTesla 1 0.0 100.0 1.000 0.02002 0.0 100.0 100.0 2.0000 3 0.0 100.0 320.0 7.3600 4 180.0 100.0 1.0000.0170 5 0.0 150.0 1.000 0.0195 6 0.0 200.0 1.000 0.0150

For the first shielded Y uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a′=−0.002 m andy=−b′=−0.056 m, respectively. Four constraint points were chosen todefine the characteristics of the field inside an imaging volume. Thefirst two constraint points define a gradient strength of 20 mT/m insidethe imaging volume with a 19% on-axis linearity. The reason for usingtwo constraint points along the gradient axis is that the on-axisnon-linearity of the gradient coil must be contained in acceptablelevels. The uniformity of the gradient field inside this imaging volumeis restricted to less than 20%from its actual value inside a 40cm Dsvvolume. This set of constraints is displayed in Table 3.

TABLE 3 Constraints for the first Y shielded Uniplanar module j X_(j) inmm Y_(j) in mm Z_(j) in mm B_(ysc) in mTesla 1 0.0 51.0 0.000 0.0200 20.0 251.0 0.000 3.240 3 0.0 51.0 200.0 0.0155 4 180.0 51.0 0.000 0.0180

For the second shielded Y uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a=−0.006 m andy=−b=−0.050 m, respectively. Four constraint points were chosen todefine the characteristics of the field inside an imaging volume. Thefirst two constraint points define a gradient strength of 20 mT/m insidethe imaging volume with a 19% on-axis linearity. The reason for usingtwo constraint points along the gradient axis is that the on-axisnon-linearity of the gradient coil must be contained in acceptablelevels. The uniformity of the gradient field inside this imaging volumeis restricted to less than 20% from its actual value inside a 40 cm DSV.This set of constraints is displayed in Table 4.

TABLE 4 Constraints for the second Y shielded Uniplanar module j X_(j)in mm Y_(j) in mm Z_(j) in mm B_(ysc) in mTesla 1 0.0 51.0 0.000 0.02002 0.0 251.0 0.000 3.240 3 0.0 51.0 200.0 0.0155 4 180.0 51.0 0.0000.0180

For the first shield z uniplanar module, by employing the streamfunction technique, the discrete current pattern for the primary coilconsists from 17 discrete loops (FIG. 4A) where each loops carries aconstant current of 216.68 Amps. Similarly, the secondary coil's currentdensity is approximated by a 10 loops (FIG. 4B) where each loop carriesa constant current of 216.68 Amps. Using the Biot-Savart law, the ycomponent of the magnetic field is evaluated along the yz plane atx=0.0, as shown in FIG. 5. Table 5 shows the electrical characteristicsfor the first module of the shielded Z uniplanar gradient coil which aredesigned for main magnets with vertically directed fields. The risetime, slew rates and gradient strengths were evaluated assuming a400V/330 A gradient amplifier.

Employing the stream function technique, the discrete current patternfor the primary coil consisted of 23 discrete loops (FIG. 6A), whereeach loop carries a constant current of 188.6 Amps. Similarly, thesecondary coil's current density can be approximated by 15 loops (FIG.6B), where each loop carries a constant current of 188.6 Amps. Using theBiot-Savart law, the y component of the magnetic field is evaluatedalong the yz plane at x=0.0, as shown in FIG. 7. Table 5 shows theelectrical characteristics for the second module of the shielded Zuniplanar gradient coil which are designed for main magnets withvertically directed fields. The rise time, slew rates and gradientstrengths were evaluated assuming a 400V/330 A gradient amplifier.

TABLE 5 Electrical Characteristics for the Two shielded Z UnipolarModules First Shielded Second Z Uniplanar Shielded Z Property moduleUniplanar module Primary plane 0.002 m 0.006 m location Shielding plane0.056 m 0.050 m location Number of 17/10 23/15 discrete loopsPrimary/Secondary Gradient Strength 30.46 mT/m 35 mT/m @ 330 A Linearityat 15.35% 17.68% z = ±20 cm Uniformity at 18.5% 20.92% y = ±20 cmInductance in μH 346 μH 567 μH Resistance in mΩ 115 mΩ 185 mΩ Rise Timein μsec 315 μsec 552 μsec Linear Slew 97 T/m/sec 63 T/m/sec Rate @ 400 VSinusoidal Slew 106 T/m/sec 76 T/m/sec Rate @ 400 V

Further, the mutual energy between these two modules was evaluated basedon Equation (32). As FIG. 8 indicates, by axially (along z) shifting thefirst module coil relative to the second one, the mutual energy, andhence the mutual inductance, of these two modules goes through zero whenthe first module is shifted by z=244 mm away from the second coilmodule. In this case, there will be no effect to the rise time or slewrate of the combined module system as long as the coils are axiallyseparated by z=244 mm relative to each other. As FIG. 8 also indicates,if the first coil module is placed on top of the second coil module, themutual energy between these two modules is 8.34 Joules which is almostcomparable with the self-energy or either one of the modules. In thiscase, driving these two coils in series will affect rise time and slewrate significantly.

For the first shield y uniplanar module, by employing the streamfunction technique, the discrete current pattern for the primary coilconsists of 15 discrete loops (FIG. 9A) where each loops carries aconstant current of 348.51 Amps. Similarly, the secondary coil's currentdensity is approximated by 10 loops (FIG. 9B) where each loop carries aconstant current of 348.51 Amps. Table 6 shows the electricalcharacteristics for the first module of the shielded Y uniplanargradient coil which is designed for main magnets with verticallydirected fields. The rise time, slew rates and gradient strengths wereevaluated assuming a 400V/330 A gradient amplifier.

Employing the stream function technique, the discrete current patternfor the second primary Y uniplanar module consists of 17 discrete loops(FIG. 10A) where each loop carries a constant current of 365.39 Amps.Similarly, the secondary coil's current density is approximated by 10loops (FIG. 10B) where each loop carries a constant current of 365.39Amps. Table 6 shows the electrical characteristics for the second moduleof the shielded Y uniplanar gradient coil which is designed for mainmagnets with vertically directed fields. The rise time, slew rates andgradient strengths were evaluated assuming a 400V/330 A gradientamplifier.

TABLE 6 Electrical Characteristics for the Two shielded Y UniplanarModules First Shielded Second Y Uniplanar Shielded Y Property moduleUniplanar module Primary plane 0.002 m 0.006 m location Shielding plane0.056 m 0.050 m location Number of 15/10 17/12 discrete loopsPrimary/Secondary Gradient Strength 19 mT/m 18 mT/m @ 330 A Linearity at21.42% 20.86% z = ±20 cm Uniformity at 20.01% 20.42% y = ±20 cmInductance in μH 162 μH 184 μH Resistance in mΩ 115 mΩ 175 mΩ Rise Timein μsec 148 μsec 177 μsec Linear Slew 129 T/m/sec 101 T/m/sec Rate @ 400V Sinusoidal Slew 141 T/m/sec 117 T/m/sec Rate @ 400 V

Further, the mutual energy between these two modules was evaluated basedon Equation (32). As FIG. 11 indicates, by axially (along z) shiftingthe first module coil relative to the second one, the mutual energy, andhence the mutual inductance, of these two modules goes through zero whenthe first module is shifted by z=258 mm away from the second coilmodule. In this case, there will be no effect on the rise time or slewrate of the combined module system as long as the coils are axiallyseparated by z=258 mm relative to each other (FIG. 11).

The non-shielded case, is represented by the above equations where b∞.It is to be appreciated that the uniplanar design may be adapted to abore-type system with a horizontally directed magnetic field. Thedevelopment and design for a horizontal system will now be presented. Inthis case, the main magnetic field is directed along the z-axis.

Considering the plane for the primary coil positioned at y=−a, while theplane for the secondary coil positioned at y=−b, and confining thecurrent density on the xz plane, the expression of the currentdistribution J(x,z) is:

{right arrow over (J)}(x,z)=[J _(x)(x,z){circumflex over (x)}+J_(z)(x,z){circumflex over (z)}]  (33)

Therefore the expressions of the two components A_(x), A_(z) of themagnetic vector potential for y≧−a are: $\begin{matrix}\begin{matrix}{A_{x} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}} \\{\quad \left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right\rbrack}\end{matrix} & (34) \\\begin{matrix}{A_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}} \\{\quad \left\lbrack {{{J_{z}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {{J_{z}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right\rbrack}\end{matrix} & (35)\end{matrix}$

In addition, the expressions of the two components A_(x), A_(z) of themagnetic vector potential for y≦−b are: $\begin{matrix}\begin{matrix}{A_{x} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}} \\{\quad \left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack}\end{matrix} & (36) \\\begin{matrix}{A_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}} \\{\quad \left\lbrack {{{J_{z}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {{J_{z}^{- b}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack}\end{matrix} & (37)\end{matrix}$

where J_((x,z))(α,β) represents the double Fourier transform of theJ_((x,z))(x,z) components of the current density, respectively.Considering the current continuity equation ({overscore (V)}·J=0), therelationship for these two components of the current density in theFourier domain is: $\begin{matrix}{{J_{z}\left( {\alpha,\beta} \right)} = {{- \frac{\alpha}{\beta}}\quad {J_{x}\left( {\alpha,\beta} \right)}}} & (38)\end{matrix}$

In this case, Equations (35) and (37) have the form: $\begin{matrix}\begin{matrix}{A_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \quad \frac{\alpha}{\beta}} \right)}}}}}} \\{\quad \left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right\rbrack} \\{\quad {{{for}\quad y} \geq {- a}}}\end{matrix} & (39) \\\begin{matrix}{A_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \quad \frac{\alpha}{\beta}} \right)}}}}}} \\{\quad \left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack} \\{\quad {{{for}\quad y} \leq {- b}}}\end{matrix} & (40)\end{matrix}$

Furthermore, the expressions of the two components for magnetic vectorpotential at the surfaces of the two planes (y=−a, and y=−b) have theform: $\begin{matrix}\begin{matrix}{A_{x} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}} \\{\quad {{\left\lbrack {{J_{x}^{- a}\left( {\alpha,\beta} \right)} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}}} \right\rbrack \quad {for}\quad y} = {- a}}}\end{matrix} & (41) \\\begin{matrix}{A_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \quad \frac{\alpha}{\beta}} \right)}}}}}} \\{\quad {{\left\lbrack {{J_{x}^{- a}\left( {\alpha,\beta} \right)} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}}} \right\rbrack \quad {for}\quad y} = {- a}}}\end{matrix} & (42) \\\begin{matrix}{A_{x} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}} \\{\quad {{\left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}} + {J_{x}^{- b}\left( {\alpha,\beta} \right)}} \right\rbrack \quad {for}\quad y} = {- b}}}\end{matrix} & (43) \\\begin{matrix}{A_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{\frac{{\alpha}{\beta}}{\sqrt{\alpha^{2} + \beta^{2}}}{^{{\quad \alpha \quad x} + {\quad \beta \quad z}}\left( {- \quad \frac{\alpha}{\beta}} \right)}}}}}} \\{\quad {{\left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({b - a})}}} + {J_{x}^{- b}\left( {\alpha,\beta} \right)}} \right\rbrack \quad {for}\quad y} = {- b}}}\end{matrix} & (44)\end{matrix}$

The expression for the component of the gradient field coincidental withthe direction of the main magnetic field is B_(z), and has the followingform:$B_{z} = \quad \left. {- {\partial_{y}A_{x}}}\Rightarrow \begin{matrix}\begin{matrix}{B_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}{{\beta }^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}}} \\{\quad \left. \left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + b})}}}} \right. \right\rbrack} \\{\quad {{{for}\quad y} \geq {- a}}}\end{matrix} & (45) \\\begin{matrix}{B_{z} = \quad {{- \frac{\mu_{0}}{8\pi^{2}}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}{{\beta }^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}}}}}} \\{\quad \left\lbrack {{{J_{x}^{- a}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + a})}}} + {{J_{x}^{- b}\left( {\alpha,\beta} \right)}^{\sqrt{\alpha^{2} + \beta^{2}}{({y + b})}}}} \right\rbrack} \\{\quad {{{for}\quad y} \leq {- b}}}\end{matrix} & (46)\end{matrix} \right.$

Considering the shielding requirements B_(z)=0 at y=−b, Equation (46)relates the current densities for the two planes as:

J _(x) ^(−b)(αβ)=−e ^({square root over (α 2 _(+β) 2 +L )}) ^((b−a)) J_(x) ^(−a)(α,β)  (47)

and Equation (45) becomes: $\begin{matrix}\begin{matrix}{B_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}{{\beta }^{{\quad \alpha \quad x} + {\quad \beta \quad z}}}^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}{J_{x}^{- a}\left( {\alpha,\beta} \right)}}}}}} \\{\quad {{\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack \quad {for}\quad y} \geq {- a}}}\end{matrix} & (48)\end{matrix}$

In addition, the stored magnetic energy of the coil is given by Equation(48a) as:$W_{m} = \left. {\frac{1}{2}{\int_{v}\quad {{^{3}x}\quad {\overset{\rightarrow}{A} \cdot \overset{\rightarrow}{J}}}}}\Rightarrow \begin{matrix}\begin{matrix}{W_{m} = \quad {\frac{\mu_{0}}{16\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}{\beta}\frac{\sqrt{\alpha^{2} + \beta^{2}}}{\beta^{2}}{{J_{x}^{- a}\left( {\alpha,\beta} \right)}}^{2}}}}}} \\{\quad \left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}\end{matrix} & \text{(48a)}\end{matrix} \right.$

In this section, the theoretical development for the transverseuniplanar shielded gradient coil will be presented. Specifically, the Ygradient coil will be presented in its entirety.

The non-shielded case is represented by the above equations, where b∞.For the Y gradient coil, the z component of the gradient field issymmetric along the z and x directions. In this case, Equation (48)becomes: $\begin{matrix}\begin{matrix}{B_{z} = \quad {\frac{\mu_{0}}{8\pi^{2}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}{\beta}\quad \cos \quad \left( {\alpha \quad x} \right)\quad \cos \quad \left( {\beta \quad z} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}}}}}} \\{\quad {{{{J_{x}^{- a}\left( {\alpha,\beta} \right)}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}\quad {for}\quad y} \geq {- a}}}\end{matrix} & (49)\end{matrix}$

which leads to the expression of the Fourier transform for the J_(x)^(−a)(α,β) as: $\begin{matrix}{{J_{x}^{- a}\left( {\alpha,\beta} \right)} = {4{\int{\int_{0}^{+ \infty}{{x}{z}\quad \cos \quad \left( {\alpha \quad x} \right)\quad \cos \quad \left( {\beta \quad z} \right){J_{x}^{- a}\left( {x,z} \right)}}}}}} & (50)\end{matrix}$

Based on the energy minimization mechanism, the functional E isconstructed in terms of the magnetic field and the stored magneticenergy as: $\begin{matrix}{{E\left( J_{x}^{- a} \right)} = {W_{m} - {\sum\limits_{j = 1}^{N}{\lambda_{j}\left\lbrack {{B_{z}\left( {\overset{\rightarrow}{r}}_{j} \right)} - {B_{z\quad {SC}}\left( {\overset{\rightarrow}{r}}_{j} \right)}} \right\rbrack}}}} & \text{(50a)}\end{matrix}$

where λ_(j) are the Lagrange multipliers, B_(z)(r_(j)) is the calculatedvalue of the magnetic field at the constraintpoints r_(j) andB_(zSC)(r_(j)) are the constraint values of the magnetic field at theconstraint points.

Minimizing E with respect to the current density J_(x) ^(−a), a matrixEquation for J_(x) ^(−a) is: $\begin{matrix}{J_{x}^{- a} = {\frac{\beta^{2}}{\sqrt{\alpha^{2} + \beta^{2}}}{\sum\limits_{j = 1}^{N}{\lambda_{j}\cos \quad \left( {\alpha \quad x_{j}} \right)\quad \cos \quad \left( {\beta \quad z_{j}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}}} & (51)\end{matrix}$

where the Lagrange multipliers λ_(j) are determined via the constraintequation: $\begin{matrix}{{\sum\limits_{j = 1}^{N}{C_{i\quad j}\lambda_{j}}} = {B_{z\quad {SC}_{i}}\quad {with}}} & \quad \\\begin{matrix}{C_{i\quad j} = \quad {\frac{\mu_{0}}{2\pi^{2}}{\int{\int_{0}^{+ \infty}{{\alpha}{\beta}\quad \cos \quad \left( {\alpha \quad x_{i}} \right)\quad \cos \quad \left( {\beta \quad z_{i}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{i} + a})}}}}}}} \\{\quad {\sum\limits_{j = 1}^{N}{\frac{\beta^{2}}{\sqrt{\alpha^{2} + \beta^{2}}}\quad \lambda_{j}\cos \quad \left( {\alpha \quad x_{j}} \right)\quad \cos \quad \left( {\beta \quad z_{j}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}} \\{\quad \left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack \quad}\end{matrix} & (52)\end{matrix}$

Upon determination of the Lagrange multipliers the continuous currentdensity distribution for J_(x) ^(−a) and J_(z) ^(−a) components of thecurrent density are generated. For the shielding coil, its currentdensity is derived by determining the current density of the primarycoil via Equation (52) and subsequently applying the inverse transformto Equation (47). Upon determination of the continuous current densitiesfor both coils, their associated discrete current patterns, which arevery close approximations for the continuous current densities, can begenerated by applying the stream function technique. In order to ensurethe integrity of the discretization process, the magnetic field isre-evaluated inside and outside the imaging volume by applying theBiot-Savart formula to both discrete current distributions.

For the X gradient coil, the z component of the gradient field must besymmetric along the z and y directions while it must be anti-symmetricalong the x direction. In this case, Equation (48) becomes:$\begin{matrix}\begin{matrix}{B_{z} = \quad {{- \frac{\mu_{0}}{8\pi^{2}}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}{\beta}\quad \sin \quad \left( {\alpha \quad x} \right)\quad \cos \quad \left( {\beta \quad z} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}}}}}} \\{\quad {{{{{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}\quad {for}\quad y} \geq {- a}}}\end{matrix} & (53)\end{matrix}$

which leads to the expression of the Fourier transform for the J_(x)^(−a)(α,β) as: $\begin{matrix}{{{J_{x}^{- a}\left( {\alpha,\beta} \right)} = {\quad {{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)}\quad {and}}}{{{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)} = {4{\int{\int_{0}^{+ \infty}{{x}{z}\quad \sin \quad \left( {\alpha \quad x} \right)\quad \cos \quad \left( {\beta \quad z} \right){J_{x}^{- a}\left( {x,y} \right)}}}}}}} & (54)\end{matrix}$

Based on the energy minimization mechanism, the functional E isconstructed in terms of the magnetic field and the stored magneticenergy as: $\begin{matrix}{{E\left( {\hat{J}}_{x}^{- a} \right)} = {W_{m} - {\sum\limits_{j = 1}^{N}{\lambda_{j}\left\lbrack {{B_{z}\left( {\overset{\rightarrow}{r}}_{j} \right)} - {B_{z\quad {SC}}\left( {\overset{\rightarrow}{r}}_{j} \right)}} \right\rbrack}}}} & \text{(54a)}\end{matrix}$

where λ_(j) are the Lagrange multipliers, B_(z)(r_(j)) is the calculatedvalue of the magnetic field at the constraint points r_(j) andB_(zSC)(r_(j)) are the constraint values of the magnetic field at theconstraint points.

Minimizing E with respect to the current density Ĵ_(x) ^(−a), the matrixequation for Ĵ_(x) ^(−a) is: $\begin{matrix}{{\hat{J}}_{x}^{- a} = {{- \quad \frac{\beta^{2}}{\sqrt{\alpha^{2} + \beta^{2}}}}{\sum\limits_{j = 1}^{N}{\lambda_{j}\sin \quad \left( {\alpha \quad x_{j}} \right)\quad \cos \quad \left( {\beta \quad z_{j}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}}} & (55)\end{matrix}$

where the Lagrange multipliers λ_(j) are determined via the constraintequation: $\begin{matrix}{{\sum\limits_{j = 1}^{N}{C_{i\quad j}\lambda_{j}}} = {B_{z\quad {SC}_{i}}\quad {with}}} & \quad \\\begin{matrix}{C_{i\quad j} = \quad {\frac{\mu_{0}}{2\pi^{2}}{\int{\int_{0}^{+ \infty}{{\alpha}{\beta}\quad \sin \quad \left( {\alpha \quad x_{i}} \right)\quad \cos \quad \left( {\beta \quad z_{i}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{i} + a})}}}}}}} \\{\quad {\sum\limits_{j = 1}^{N}{\frac{\beta^{2}}{\sqrt{\alpha^{2} + \beta^{2}}}\quad \lambda_{j}\sin \quad \left( {\alpha \quad x_{j}} \right)\quad \cos \quad \left( {\beta \quad z_{j}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}} \\{\quad \left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack \quad}\end{matrix} & (56)\end{matrix}$

Upon determination of the Lagrange multipliers the continuous currentdensity distribution for J_(x) ^(−a) and J_(z) ^(−a) components of thecurrent density are generated. For the shielding coil, its currentdensity can be derived by determining the current density of the primarycoil via Equation (56) and subsequently applying the inverse transformto Equation (47). Upon determination of the continuous current densitiesfor both coils, their associated discrete current patterns which is avery close approximation for the continuous current density can begenerated by applying the stream function technique. In order to ensurethe integrity of the discretization process, the magnetic field isre-evaluated inside and outside the imaging volume by applying theBiot-Savart formula to both discrete current distributions.

In this section, the theoretical development for the axial uniplanarshielded gradient coil will be presented. For the Z gradient coil, the zcomponent of the gradient field must be symmetric along the x and ydirections. In this case, Equation (48) becomes: $\begin{matrix}\begin{matrix}{B_{z} = \quad {{- \frac{\mu_{0}}{8\pi^{2}}}{\int{\int_{- \infty}^{+ \infty}{{\alpha}{\beta}\quad \cos \quad \left( {\alpha \quad x} \right)\quad \sin \quad \left( {\beta \quad z} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y + a})}}}}}}} \\{\quad {{{{{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)}\left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack}\quad {for}\quad y} \geq {- a}}}\end{matrix} & (57)\end{matrix}$

which leads to the expression of the Fourier transform for the J_(x)^(−a) (α,β) as: $\begin{matrix}{{{J_{x}^{- a}\left( {\alpha,\beta} \right)} = {\quad {{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)}\quad {and}}}{{{\hat{J}}_{x}^{- a}\left( {\alpha,\beta} \right)} = {4{\int{\int_{0}^{+ \infty}{{x}{z}\quad \cos \quad \left( {\alpha \quad x} \right)\quad \sin \quad \left( {\beta \quad z} \right){J_{x}^{- a}\left( {x,z} \right)}}}}}}} & (58)\end{matrix}$

Based on the energy minimization mechanism, the functional E isconstructed in terms of the magnetic field and the stored magneticenergy as: $\begin{matrix}{{E\left( {\hat{J}}_{x}^{- a} \right)} = {W_{m} - {\sum\limits_{j = 1}^{N}{\lambda_{j}\left\lbrack {{B_{z}\left( {\overset{\rightarrow}{r}}_{j} \right)} - {B_{z\quad {SC}}\left( {\overset{\rightarrow}{r}}_{j} \right)}} \right\rbrack}}}} & \text{(58a)}\end{matrix}$

where λ_(j) are the Lagrange multipliers, B_(z)(r_(j)) is the calculatedvalue of the magnetic field at the constraint points r_(j) andB_(zSC)(r_(j)) are the constraint values of the magnetic field at theconstraint points.

Minimizing E with respect to the current density Ĵ_(x) ^(−a), the matrixequation for Ĵ_(x) ^(−a) is: $\begin{matrix}{{\hat{J}}_{x}^{- a} = {{- \quad \frac{\beta^{2}}{\sqrt{\alpha^{2} + \beta^{2}}}}{\sum\limits_{j = 1}^{N}{\lambda_{j}\cos \quad \left( {\alpha \quad x_{j}} \right)\quad \sin \quad \left( {\beta \quad z_{j}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}}} & (59)\end{matrix}$

where the Lagrange multipliers λ_(j) are determined via the constraintEquation: $\begin{matrix}{{\sum\limits_{j = 1}^{N}{C_{i\quad j}\lambda_{j}}} = {B_{z\quad {SC}_{i}}\quad {with}}} & \quad \\\begin{matrix}{C_{i\quad j} = \quad {\frac{\mu_{0}}{2\pi^{2}}{\int{\int_{0}^{+ \infty}{{\alpha}{\beta}\quad \cos \quad \left( {\alpha \quad x_{i}} \right)\quad \sin \quad \left( {\beta \quad z_{i}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{i} + a})}}}}}}} \\{\quad {\sum\limits_{j = 1}^{N}{\frac{\beta^{2}}{\sqrt{\alpha^{2} + \beta^{2}}}\quad \lambda_{j}\cos \quad \left( {\alpha \quad x_{j}} \right)\quad \sin \quad \left( {\beta \quad z_{j}} \right)^{{- \sqrt{\alpha^{2} + \beta^{2}}}{({y_{j} + a})}}}}} \\{\quad \left\lbrack {1 - ^{{- 2}\sqrt{\alpha^{2} + \beta^{2}}{({b - a})}}} \right\rbrack \quad}\end{matrix} & (60)\end{matrix}$

Upon determination of the Lagrange multipliers, the continuous currentdensity distribution for J_(x) ^(−a) and J_(z) ^(−a) components of thecurrent density are generated. For the shielding coil, its currentdensity can be derived by determining the current density of the primarycoil via Equation (60) and subsequently applying the inverse transformto Equation (47). Upon determination of the continuous current densitiesfor both coils, their associated discrete current patterns, which arevery close approximations of the continuous current density, can begenerated by applying the stream function technique. In order to ensurethe integrity of the discretization process, the magnetic field isre-evaluated inside and outside the imaging volume by applying theBiot-Savart formula to both discrete current distributions.

For the first shielded Z uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a′=−0.006 m andy=−b′=−0.03170 m, respectively. Five constraint points define thecharacteristics of the field inside an imaging volume. The first threeconstraint points define a gradient strength of 20 mT/m inside theimaging volume with a 9% on-axis linearity. Three constraint pointsalong the gradient axis are used to contain the on-axis non-linearity ofthe gradient coil in acceptable levels. The uniformity of the gradientfield inside this imaging volume is restricted to less than 15% from itsactual value. This set of constraints is displayed in Table 7.

TABLE 7 Constraints for the first Z shielded Uniplanar module j X_(j) inmm Y_(j) in mm Z_(j) in mm B_(zsc) in mTesla 1 0.0 0.000 1.000 0.0200 20.0 0.000 100.0 2.0000 3 0.0 0.000 320.0 5.8400 4 180.0 0.000 1.0000.0190 5 0.0 200.0 1.000 0.0170

For the second shielded Z uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a=−0.003 m andy=−b=−0.03355 m, respectively. Five constraint points define thecharacteristics of the field inside an imaging volume. The first threeconstraint points define a gradient strength of 30 mT/m inside theimaging volume with a 15% on-axis linearity. Three constraint pointsalong the gradient axis are used to contain the on-axis non-linearity ofthe gradient coil in acceptable levels. The uniformity of the gradientfield inside this imaging volume is restricted to less than 27% from itsactual value. This set of constraints is displayed in Table 8.

TABLE 8 Constraints for the second Z shielded Uniplanar module j X_(j)in mm Y_(j) in mm Z_(j) in mm B_(zsc) in mTesla 1 0.0 0.0 1.000 0.0300 20.0 0.0 100.0 3.0000 3 0.0 0.0 320.0 8.1400 4 180.0 0.0 1.000 0.0290 50.0 200.0 1.000 0.0220

For the first shielded Y uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a′=−0.006 m andy=−b′=−0.03272 m, respectively. Four constraint points define thecharacteristics of the field inside an imaging volume. The first twoconstraint points define a gradient strength of 20 mT/m inside theimaging volume with a 5% on-axis linearity. Two constraint points alongthe gradient axis are used to contain the on-axis non-linearity of thegradient coil in acceptable levels. The uniformity of the gradient fieldinside this imaging volume is restricted to less than 22.5% from itsactual value inside a 40 cm DSV volume. This set of constraints isdisplayed in Table 9.

TABLE 9 Constraints for the first Y shielded Uniplanar module j X_(j) inmm Y_(j) in mm Z_(j) in mm B_(zsc) in mTesla 1 0.0 1.0 0.000 0.0200 20.0 180.0 0.000 3.440 3 0.0 1.0 200.0 0.0155 4 180.0 1.0 0.000 0.0180

For the second shielded Y uniplanar module, the positions of the twoplanes for the primary and secondary coils are at y=−a=−0.003 m andy=−b=−0.03712 m, respectively. Four constraint points define thecharacteristics of the field inside an imaging volume. The first twoconstraint points define a gradient strength of 30 mT/m inside theimaging volume with a 6% on-axis linearity. Two constraint points alongthe gradient axis are used to contain the on-axis non-linearity of thegradient coil in acceptable levels. The uniformity of the gradient fieldinside this imaging volume is restricted to less than 22% from itsactual value inside a 40 cm DSV. This set of constraints is displayed inTable 10.

TABLE 10 Constraints for the second Y shielded Uniplanar module j X_(j)in mm Y_(j) in mm Z_(j) in mm B_(zsc) in mTesla 1 0.0 1.0 0.000 0.0300 20.0 150.0 0.000 4.240 3 0.0 1.0 200.0 0.0235 4 180.0 1.0 0.000 0.0260

For the first shield z uniplanar module, by employing the streamfunction technique, the discrete current pattern for the primary coil iscomprised of 22 positive and 3 negative discrete loops (FIG. 12A) whereeach loop carries a constant current of 211.43 Amps. Similarly, thesecondary coil's current density is approximated by 13 loops (FIG. 12B)where each loop carries a constant current of 211.43 Amps. Table 11shows the electrical characteristics for the first module of theshielded Z uniplanar gradient coil which is designed for main magnetswith horizontally directed fields. The rise time, slew rates andgradient strengths were evaluated assuming a 400V/330 A gradientamplifier.

For the second shield Z uniplanar module the stream function techniqueleads to a discrete current pattern for the primary coil comprised of 21discrete positive and 3 negative loops (FIG. 13A) where each loopcarries a constant current of 248.6 Amps. Similarly, the secondarycoil's current density is approximated by 9 loops (FIG. 13B) where eachloop carries a constant current of 248.6 Amps. Table 11 shows theelectrical characteristics for the second module of the shielded Zuniplanar gradient coil which is designed for main magnets withhorizontally directed fields. The rise time, slew rates and gradientstrengths were evaluated assuming a 400V/330 A gradient amplifier.

TABLE 11 Electrical Characteristics for the two shielded Z UniplanarModules First Shielded Second Z Uniplanar Shielded Z Property moduleUniplanar module Primary plane 0.006 m 0.003 m location Shielding plane0.0321 m 0.03355 m location Number of 22/13 21/9 discrete loopsPrimary/Secondary Gradient Strength 31.2 mT/m 39.8 mT/m @ 330 ALinearity at 5.68% 6.96% z = ±20 cm Uniformity at 21.76% 28.4% y = ±20cm Inductance in μH 190 μH 191 μH Resistance in mΩ 145 mΩ 105 mΩ RiseTime in μsec 178 μsec 173 μsec Linear Slew 175 T/m/sec 230 T/m/sec Rate@ 400 V Sinusoidal Slew 198 T/m/sec 252 T/m/sec Rate @ 400 V

Further, the mutual energy between the first and second modules wasevaluated based on Equation (32). As FIG. 14 indicates, by axiallyshifting (along z) the first module coil relative to the second one, themutual energy, and hence, the mutual inductance of these two modules,goes through zero when the first module is shifted by z=140 mm from thesecond coil module. In this case, there will be no effect on the risetime or slew rate of the combined module system as long as the coils areaxially separated by z=140 mm relative to each other. As FIG. 14 alsoindicates, if the first coil module is placed on top of the second coilmodule, the mutual energy between these two modules is 6.54 Joules,which is comparable with the self-energy of either one of the modules.Driving these two coils in series has significant effects on rise timeand slew rate.

For the first shield Y uniplanar module, by employing the streamfunction technique, the discrete current pattern for the primary coil iscomprised of 21 discrete loops (FIG. 15A) where each loop carries aconstant current of 223.81 Amps. Similarly, the secondary coil's currentdensity is approximated by 16 loops (FIG. 15B) where each loop carries aconstant current of 223.81 Amps. Table 12 shows the electricalcharacteristics for the first module of the shielded Y uniplanargradient coil, which is designed for main magnets with horizontallydirected fields. The rise time, slew rates and gradient strengths areevaluated assuming a 400V/330 A gradient amplifier.

For the second Y shielded uniplanar module, the stream functiontechnique leads to a discrete current pattern for the primary coil iscomprised of 15 discrete loops (FIG. 16A) where each loop carries aconstant current of 232.85 Amps. Similarly, the secondary coil's currentdensity is approximated by 10 loops (FIG. 16B) where each loop carries aconstant current of 232.85 Amps. Table 12 shows the electricalcharacteristics for the second module of the shielded Y uniplanargradient coil, which is designed for main magnets with horizontallydirected fields. The rise time, slew rates and gradient strengths areevaluated assuming a 400V/330 A gradient amplifier.

TABLE 12 Electrical Characteristics for the two shielded Y UniplanarModules First Shielded Y Uniplanar Second Shielded Y Property moduleUniplanar module Primary plane 0.006 m 0.003 m location Shielding plane0.03272 m 0.03712 m location Number of 21/16 15/10 discrete loopsPrimary/Secondary Gradient Strength 29.5 mT/m 42.5 mT/m @ 330 ALinearity at 5.86% 6.24% y = ±20 cm Uniformity at 23.71% 23.12% z = ±20cm Inductance in μH 370 μH 210 μH Resistance in mΩ 155 mΩ 125 mΩ RiseTime in μsec 350 μsec 193 μsec Linear Slew 85 T/m/sec 220 T/m/sec Rate @400 V Sinusoidal Slew 95 T/m/sec 265 T/m/sec Rate @ 400 V

Further, the mutual energy between these two modules was evaluated basedon Equation (32). As FIG. 17 indicates, by axially shifting (along z)the first module coil relative to the second one, the mutual energy, andhence the mutual inductance, of these two modules goes through zero whenthe first module is shifted by z=296 mm away from the second coilmodule. In this case, there is no effect on the rise time or slew rateof the combined module system as long as the coils are axially separatedby z=296 mm relative to each other (FIG. 17).

It should be appreciated that the specified current patterns can bechanged to produce either better linearity at the price of coilefficiency, and/or greater efficiency at the price of linearity.Further, the dimensions of the gradient coils can be changed to beincreased or decreased according to the preferred application. Inaddition, the lengths of the primary coils and/or the secondary coilscan be similar or different.

The present invention is applicable to other alternative gradient coilgeometries, such as elliptical, bi-planar, flared, etc., as well as theasymmetric gradient coil designs or any combination thereof. The presentinvention is also applicable to the design of gradient coil structuressuitable for horizontally oriented or closed magnet systems. Further,the disclosed primary and screen coil set can be bunched (concentrated)or thumbprint designs generated using forward or inverse approachmethods. In addition, the primary and the shield coils can have anypossible mixing of bunched and/or thumbprint designs. It is contemplatedthat zero net thrust force or torque can be incorporated into theproposed design algorithm in a known manner.

With reference to FIGS. 18A, 18B, and 18C, alternate ways ofelectrically connecting the first and second uniplanar gradient coilsets are provided. These methods include series and parallel connectionsof the two uniplanar gradient coil sets in such ways that one or bothgradient coil sets may be selectively excited depending on theparticular application. For example, in FIG. 18A, the first gradientcoil set 130 and second gradient coil set 132 are electrically connectedsuch that with the first and second switches 134, 136 both in position1, only the first gradient coil set is excited. In contrast, with bothswitches 134, 136 in position 2, both the first and second gradient coilsets 130, 132 are excited while connected in series.

Alternately, in FIG. 18B, the first gradient coil set 130 is selectivelyexcited with the first switch 140 in the “on position” and the secondswitch 142 in the “off position.” Conversely, the second gradient coilset 132 is selectively excited with the second switch 142 in the onposition and the first switch 140 in the off position. With bothswitches 140, 142 in the on position, both the first and second gradientcoil sets 130, 132 are excited while connected in parallel.

The invention has been described with reference to the preferredembodiments. Obviously, modifications and alterations will occur toothers upon reading and understanding the preceding detaileddescription. It is intended that the invention be construed as includingall such modifications and alterations insofar as they come within thescope of the appended claims or the equivalents thereof.

Having thus described the preferred embodiments, the invention is nowclaimed to be:
 1. A magnetic resonance imaging apparatus comprising: amain magnet which defines an examination region in which the main magnetgenerates a main magnetic field; a couch for supporting a subject withinthe examination region; a planar gradient coil assembly disposed atleast on one side of the subject for generating gradient magnetic fieldsacross the examination region, the planar gradient coil assemblyincluding: at least a first primary planar gradient coil set and asecond primary planar gradient coil set disposed in an overlappingrelationship, the first primary planar gradient coil set being displacedrelative to the second primary planar gradient coil set such that mutualinductance between the first and second planar primary gradient coilsets is minimized; a current supply for supplying electrical current tothe planar gradient coil assembly such that magnetic field gradients areselectively generated across the examination region in the main magneticfield by the planar gradient coil assembly; a radio frequency pulsegenerator for selectively exciting magnetic resonance dipoles disposedwithin the examination region; a receiver for receiving magneticresonance signals from resonating dipoles within the examination region;and a reconstruction processor for reconstructing an imagerepresentation from the magnetic resonance signals.
 2. The magneticresonance imaging apparatus according to claim 1, wherein the planargradient coil assembly includes: at least two shield coil sets, eachshield coil set being disposed between a corresponding planar gradientcoil set and adjacent structures that are on a common side of thesubject as the planar gradient coil assembly.
 3. The magnetic resonanceimaging apparatus according to claim 2, wherein the main magnet includesa pair of pole faces which define therebetween the examination region inwhich the main magnet generates the main magnetic field vertically. 4.The magnetic resonance imaging apparatus according to claim 2, whereinthe planar gradient coil assembly resides in a plane orthogonal to themain magnetic field.
 5. The magnetic resonance imaging apparatusaccording to claim 2, wherein the main magnet is a cylindrically shapedsolenoid having a central bore which defines the examination region,said main magnet generating a horizontally directed main magnetic field.6. The magnetic resonance imaging apparatus according to claim 5,wherein the planar gradient coil assembly resides in a plane parallel tothe main magnetic field.
 7. The magnetic resonance imaging apparatusaccording to claim 1, wherein the first and second primary uniplanargradient coil sets each include three gradient coil arrays forgenerating magnetic field gradients along three orthogonal axes, eachhaving a geometric center and sweet spot in which the magnetic field itgenerates is substantially linear, said sweet spots of each primarygradient coil set being coincident with each other.
 8. The magneticresonance imaging apparatus according to claim 1, wherein the first andsecond primary uniplanar gradient coil sets each include three gradientcoil arrays for generating magnetic field gradients along threeorthogonal axes, each having a geometric center and sweet spot in whichthe magnetic field it generates is substantially linear, the sweet spotsbeing adjacent each other such that the examination region is enlarged.9. The magnetic resonance imaging apparatus according to claim 1,wherein the first and second primary gradient coil sets generategradient magnetic fields across the examination region, which gradientmagnetic fields have non-zero first derivatives in and adjacent theexamination region.
 10. The magnetic resonance imaging apparatusaccording to claim 1, wherein the planar gradient coil assembly isuniplanar and housed within the couch.
 11. The magnetic resonanceimaging apparatus according to claim 1, wherein the first primaryuniplanar gradient coil set is a high-efficiency primary gradient coilset which enhances gradient switching speeds for ultra fast MRsequencing, and the second primary gradient coil set is a low-efficiencyprimary gradient coil set having a high-quality gradient field with aslower switching speed.
 12. The magnetic resonance imaging apparatusaccording to claim 1, wherein the planar gradient coil assembly includesswitching means for selectively coupling said first and second primaryuniplanar gradient coil sets to operate in a selected one of modes inwhich (i) the first primary planar gradient coil set is energized alone,(ii) the second primary planar gradient coil set is energized alone, and(iii) both primary planar gradient coil sets are energized concurrently.13. The magnetic resonance imaging apparatus according to claim 1,further including: a second uniplanar gradient coil assembly disposedparallel to the first planar gradient coil assembly on an opposite sideof the subject.
 14. A method of magnetic resonance imaging comprising:(a) generating a vertical main magnetic field in and through anexamination region; (b) applying a first gradient magnetic field acrossthe examination region with a first planar gradient coil duringresonance excitation; (c) applying a second gradient magnetic fieldacross the examination region with a second planar gradient coil duringresonance data acquisition; and (d) acquiring and reconstructingresonance data into an image representation.
 15. The magnetic resonanceimaging method according to claim 14 further including: switching thesecond gradient magnetic field faster than the first gradient magneticfield; and the first gradient magnetic field being more linear than thesecond gradient magnetic field.
 16. The magnetic resonance imagingmethod according to claim 15 wherein the first and second gradientmagnetic fields are applied during an echo planar imaging sequence andthe second gradient magnetic fields include a series of read gradients.17. The magnetic resonance imaging method according to claim 14 furtherincluding: applying the first gradient magnetic field across a firstportion of the examination region; and, applying the second gradientmagnetic field across a second portion of the examination region, thefirst and second examination region portions being adjacent each othersuch that an elongated area is imaged.
 18. A phased array planargradient coil assembly for generating magnetic gradients across a mainmagnetic field of a magnetic resonance apparatus, the planar gradientcoil assembly comprising: a first primary planar coil set includingfirst planar x, y, and z-gradient coils, each first planar gradient coilhaving a sweet spot in which the magnetic field gradient generated issubstantially linear, the sweet spots being coincident and in anexamination region; at least a second primary planar coil set includingsecond planar x, y, and z-gradient coils, each second planar gradientcoil having a sweet spot in which the magnetic field gradient generatedis substantially linear, the second coil set sweet spots beingcoincident with each other and in the examination region; the first andsecond primary planar coil sets being disposed in an overlappingrelationship, the first primary planar coil set being displaced relativeto the second primary planar coil set such that the mutual inductancebetween the first and second planar primary coil sets is minimized; andat least one planar shielding coil set between said first and secondprimary planar coil sets and adjacent structures for reducing thegenerated magnetic field gradients to substantially zero outside of theshielding coil set.
 19. The phased array planar gradient coil assemblyaccording to claim 18, wherein the first planar x, y and z-gradientcoils have a first geometric center and the second x, y, and z-gradientcoils have a second geometric center, at least one of the first andsecond primary coil sets having x, y, and z-gradient sweet spotscoincident with each other, the examination region, the respectivegeometric center.
 20. The phased array planar gradient coil assemblyaccording to claim 18, wherein the first primary planar coil set is ahigh-efficiency primary coil set that enhances the performance of ultrafast gradient switching in MR sequences, and the second primary planarcoil set is a low-efficiency primary gradient coil set having ahigh-quality gradient field.
 21. The phased array planar gradient coilassembly according to claim 18, wherein the first primary planar coilset generates gradient magnetic fields across the examination regionwhich are more precise and with slower switching speeds than the secondprimary planar coil set, and the second primary planar coil setgenerates gradient magnetic fields which are less precise and with ahigher switching speed than the first primary planar coil set.
 22. Thephased array planar gradient coil assembly according to claim 18,further comprising: third and fourth primary planar coil sets beingdisposed in an overlapping relationship, the third primary planar coilset being displaced relative to the fourth primary planar coil set suchthat the mutual inductance between the third and fourth primary coilsets is minimized; the third and fourth primary planar coil sets beingdisposed parallel to the first and second primary planar coil sets andon an opposite side of the examination region from the first and secondplanar coil sets.
 23. The phased array planar gradient coil assemblyaccording to claim 18, wherein at least one of the first coil set sweetspot is defined asymmetrically relative to the first x,y, and z gradientcoils.
 24. The phased array planar gradient coil assembly according toclaim 18, wherein the uniplanar gradient coil assembly is housed withinan interior of a couch which supports a subject within the examinationregion.
 25. A method of designing a phased array planar gradient coilassembly for magnetic resonance imaging systems, the method comprising:(a) selecting geometric configurations for a first primary planar coilset having a corresponding first shield coil set and at least a secondprimary planar coil set having a corresponding second shield coil set;(b) generating first and second continuous current distributions for thefirst primary and shield coil sets; (c) generating third and fourthcontinuous current distributions for the second primary and shield coilsets; (d) optimizing the first primary planar coil set with the firstshield coil set utilizing an energy/inductance minimization algorithm;(e) optimizing the second primary planar coil set with the second shieldcoil set utilizing an energy/inductance minimization algorithm; (f)evaluating eddy currents within a prescribed imaging volume for both thefirst and second primary planar coil sets; (g) modifying at least onecharacteristic of the geometric configuration defined in step (a), andrepeating steps (d) through (g) when the eddy currents from either (i)the first primary and shield coil sets or (ii) the second primary andshield coil sets do not meet an eddy current target value for theprescribed imaging volume; (h) discretizing the first primary and shieldcoil sets and the second primary and shield coil sets; and (i)displacing the first primary planar coil set at least one of axially andradially relative to the second primary planar coil set such that mutualinductance between the two is minimized.
 26. The method according toclaim 25, where at least one of steps (d) and (e) include: (j)minimizing net torque for the respective primary and shield coil sets.27. The method according to claim 25, where step (g) includes: (k)modifying at least one of a length and a radius of (i) the first primaryand shield coil sets or (ii) the second primary and shield coil sets.28. The method according to claim 25, where step (g) includes: (l)modifying a field constraint of one of (i) the first primary and shieldcoil sets and (ii) the second primary and shield coil sets.
 29. Themethod according to claim 25, where step (i) includes performing amutual inductance minimization algorithm to minimize the mutualinductance between the first and second primary coil sets.
 30. Ashielded uniplanar gradient coil assembly designed by the method ofclaim 25.